Performance evaluation of the generalized type-II hybrid ARQ scheme ...
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Performance Evaluation of the Generalized Type-I1 Hybrid ARQ Scheme with Noisy Feedback on Markov Channels Vitalice K. Oduol and Salvatore D. Morgera, Fellow, ZEEE
Abstract-By taking a particular look at the generalized TypeI1 hybrid ARQ scheme on Markov channels, this work investigates the effect of feedback channel errors on the performance of ARQ systems. In the analysis of ARQ methods, it is quite often assumed that the feedback channel is error-free. However, taking into account the feedback channel errors provides a more realistic evaluation of the performance. The main contribution of this work is that it demonstrates that it is indeed possible to derive expressions for certain critical performance parameters, such as the throughput efficiency, the probabilities of packet loss, undetected error, and correct delivery. To provide a means of comparison, the paper provides, in the Appendix, a parallel set of expressions under the usual assumption of an error-free feedback channel. By use of simulations, the ARQ system performance is examined under the two cases-noiseless feedback and noisy feedback It is found that feedback channel noise can result in the loss of packets, an increase in the number of undetected errors, and the occurrence of unnecessary transmissions. To enhance the performance of the GH-I1 ARQ scheme, a predictor is used and found to provide remarkable performance improvement-lowering the probability of undetected error, reducing the number of unnecessary transmissions, and increasing the throughput efficiency. The results presented are felt to be important for system design employing ARQ error control methods.
CEC is designed in such a way that its generator matrix can be partitioned into m subblocks, each having dimension (nl x nl). The integer m is called the depth of the code used [l],[ 2 ] .In this paper, we present the scheme with block codes; the use of convolutional codes in similar ARQ systems can be found in the work of Kallel and Haccoun [3]-[5]. On channels where the error bursts (of maximum length B ) are separated by a gap (of minimum length G), generalized burst trapping [6], [7] techniques can be used. These techniques differ markedly from the GH-I1 ARQ scheme described here, in that they require certain statistical properties in the bursts, requirements that are not needed in ARQ systems [8]. For GH-I1 ARQ, the generator matrix G of the error correcting code can be partitioned into subblocks G,, to give G = [GllG21G31 . . . IGm].
For the code CECto be useful for adaptive error correction, it is designed so that the subcode C,, with the generator matrix G(') given by
G(') = [GllG21G31 . . . IG,] I. INTRODUCTION
T
HIS section presents the generalized Type-I1 hybrid ARQ scheme (GH-I1 ARQ), which is a method that employs variable redundancy transmission. The amount of redundancy is increased as the number of retransmission requests increases. So far this method of error control has been analyzed under the assumption of an error-free feedback channel. The presentation given in this work treats both cases-noisy feedback and noiseless feedback.
(1)
z 5 m,
(2)
has minimum distance d, such that d, < d, for 1 5 a < J 5 m. The code C, is the final code CECitself. The information block d is formed from an original message D based on the code CED.The mnl-bit codeword is formed from d based on CEC.This codeword is represented as c = [ClIC2(C31 . . '
Icml
(3)
where each c, corresponds to G,, i = 1 , 2 , . . . , m. The block d can be recovered from c, if and only if G, is invertible. The sequence of transmitted blocks until successful decoding A . The Generalized Type-II ARQ Scheme is c1, c2, c3, . . . ,k.When c, is received, the receiver inverts In the GH-I1 ARQ scheme two codes are used. One is a it and determines if there are any errors in the result. If errors high-rate code CED,which is used for error detection only are found, c, is combined with previously received blocks for (used as the outer code), and the other is a rate-l/m code CEC decoding followed by error detection. If errors are found, a (the inner code), which is used adaptively for error correction. NACK is sent to the transmitter to request the block c,+l, and the procedure continues. If no errors are found, an ACK Paper approved by the Editor for Coding and Applications of the IEEE Communications Society. Manuscript received April 19, 1990; revised Septem- is sent to the transmitter and the next packet is transmitted. ber 6 , 1991. This work was supported by Canada Natural Sciences and With the sequence of blocks above, the receiver performs Engineering Research Council under Grants A0912 and A6824, and Quebec error correction based on the codes C1, C2, . . . , C,, having FCAR under Grant EQ-350. minimum distances d l , da, . . . , d,. With each retransmission, V. K. Oduol is with MPB Technologies, Inc., Pointe Claire, P. Q. H9R 1E9, Canada. therefore, a code of a larger minimum distance is used for error S. D. Morgera is with the Department of Electrical Engineering, McGill correction until the code C, is reached. It is clear from this University, MontrCal, P.Q. description that the type-I1 ARQ scheme is a special case of the IEEE Log Number 9206074. 009&6778/93$03 00 0 1993 IEEE
~
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-I1 HYBRID ARQ
33
BCH
ecode
eedback Channel
Fig. 1
The GH-I1 ARQ scheme.
GH-II ARQ scheme, since it is obtained here by setting m. = 2 . We consider the use of KM codes in the GH-II ARQ scheme [2]. The generator matrix of a KM code is block structured as
M = [MllM2). . . IMm].
(4)
The matrices Gi GY1 are related to Mi as
Fig. 2. State transition from one transmission to the next
11. PERFORMANCE EVALUATION In this section, it is assumed that the error process in the channel can be taken as a Markov chain. Specifically, let The generator matrix G is then formed as G = e k ( N ) be the number of errors in a block of length N bits [ G I I G .~. .~IGm].The code with generator matrix [Gl(G21 at time IC. The sequence el(N),e2(N),e3(N), . . . , is taken . . . IG,] is a depth-i code. For an information vector d, if as a Markov chain. The way in which this is an adequate B(’) = dG,, then d can be recovered from I?(’) by the representation of a real channel is discussed in [14], where inversion process d = B(’)GL1. it is stated that the autocovariance of the process can be A block diagram of the GH-11 ARQ scheme is shown in estimated and used to determine whether a Markov chain Fig. 1, from which it is evident that the method uses code model is appropriate. If, for example, the autocovariance is an concatenation, a topic that has received considerable attention. exponentially decaying function of the lag, the Markov model Code concatenation was first proposed by Forney [9] as a can be used. In the same reference, it is stated that such forms practical technique for implementing a code with a very of the autocovariance function have been found to represent long length and a higher error correcting power. More recent “many random phenomena encountered in practice.” With this treatment of this topic can be found in Kasami and others [lo]. assumption, we can derive the expressions for the probabilThis is further discussed by Mokrani and Solimani [ l l ] , who ities of correct delivery, undetected error, and packet loss. point out that as the error rate in a channel increases, a longer, Expressions for noiseless feedback are found in the Appendix. more powerful code is needed, but that since the longer code Let C k , Dk, and Ek, be the events, that the errors at depth-k may have a greater encodeddecoder complexity, this creates are, respectively, correctable, detectable (but not correctable), doubt as to whether there would be any gain. They suggest (in and undetectable. Further, let T and F be the events, respecagreement with Forney) that code concatenation can provide tively, that the feedback message is received as true or false. the required error correcting power with a lower complexity. A packet transmitted may be retransmitted. The decision to This fact is also stated by Clark and Cain [12]. Kl0ve and retransmit is made after the corresponding feedback message Miller [13] discuss the conditions that minimum distances has been received. The feedback message can be true ( T ) of the codes must satisfy for concatenation to bring about or false (6’). The reception of a packet can result in one of an improvement. Therefore, good codes cannot be arbitrarily three possibilities. The errors may be correctable, detectable, concatenated. We mention in passing that the codes used in or undetectable. Referring to Fig. 2, the packet may end in this work do satisfy the Klove-Miller condition. one of the circled states, C4T, C3T, etc.
34
D
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 1, JANUARY 1993
F I I T~
D2F
] epth-3
k-5
Fig. 3.
j=0
Change of depth for the GH-I1 ARQ scheme.
where A. Correct Delivery
P = P(F/F),
As the transmission proceeds, the depth changes in some cases, while in others it does not change. For example, when D1 is followed by a false feedback ( F ) , the transmission of a new packet begins. The transmission of a packet begins at depth-1. Therefore, there is no depth change when 0 1 is followed by a false feedback ( F ) . If D k is followed by a true feedback ( T ) , however, the depth increases, since the requested transmission is used in a higher depth (Dk+l) upon its arrival. This action of depth change is summarized in the illustration of Fig. 3, which shows the changes until the final depth (depth-4) is reached. The labels on the arrows in Fig. 3 indicate the events causing the changes. From this figure, it can be seen that extension to higher depth systems is quite possible. This can be done by appropriately modifying the transitions from depth4. By following the transitions in Fig. 3, it can be shown [15] that the probability P J k ) of correct delivery at the kth transmission is given by
S = P ( D 1 ) .P ( T / D 1 ) .P ( F / T ) ; = P(D1) . P ( T / D 1 ) . P ( E 4 / T ) and.
P,(k) = where
P(C1)P ( T / C l ) k = l P(Dl)P(T/Dl)P2 k=2 P(Dl)P(T/D1)03aZ k=3 P ( D ~ ) P ( T / D I ) P ~ W ~ , "2- 4, ~
{
(6)
a2 = P ( D Z / T ) P ( T / D Z ) = P ( D z / T ). P ( T / D Z ) P ( D 3 / T ). P ( T / D 3 ) a4 = P ( D 4 / T ) .P ( T / D 4 ) and ,& = P ( C k / T ) .P ( T / C k ) '
a3
'
The computation of the conditional probabilities is described in Section E of the Appendix. B. Incorrect Delivery
Incorrect delivery is the delivery of a packet with undetected errors. By using Fig. 3, the events that lead to incorrect delivery are traced to give the probability P p ( k )of incorrect delivery at the k transmission of a packet as
Pf = P ( D 1 ) . P ( F / D 1 ) . This expression will be used later in determining the average probability of incorrect delivery. For now, we proceed to state a similar expression for the loss of a packet. C. Packet Loss
A condition necessary for the loss of a packet is the detection of errors in packet. This by itself is not sufficient to cause the loss of a packet, but if this is followed by a false feedback message then a packet will be lost if the subsequent feedback message is true. Thus, the loss of a packet occurs whenever the event sequence is [Errors Detected] [False feedback message] [True feedback message]. So in deriving the expressions for the loss of a packet, we will trace the event sequences that end with the sequence DFT. The loss occurs in this case for the following reason. If the feedback message following error detection is false, the transmitter will start the transmission of the next packet instead of a retransmission. The receiver will combine the two transmissions for decoding, and since these transmissions belong to different packets, the decoding will be meaningless. The receiver will not be able to detect the problem, since the equivalent number of errors will be greater than the error detecting power of the code used. A delivery will be made to the user and the transmitter informed accordingly. If this message reaches the transmitter unaltered (true feedback), the transmitter will begin transmitting the next packet. So two things have happened here. A packet has been delivered in error, and another has been lost. The scenario described here is illustrated in Fig. 4. By tracing the event sequences ending with the sequence DFT, the probability P / ( k )of packet loss at the kth transmission is given by
35
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-I1 HYBRID ARQ
Transmitter
Packets
Receover
Subblocks A 4 A3 A2 A I
€34 B3 B2
B1
c4 c3 c2 c1
Fig. 4. The loss of a packet.
in the above definitions, the following sums result:
We are now in a position to derive the expressions for the average probabilities of correct delivery and undetected error. D. Average Probability of Undetected Error and Correct Delivery Suppose we have P packets to transmit. We define the probabilities of undetected error and of correct delivery as the limits
When evaluated, (9) and (10) are obtained as shown at the bottom of the page where w = P ( D l ) P ( T / D i ) a s w Q, 2 =
Pfp
+ so + S n 2 ,
4
S, =
4
P ~ ( I carid ) . S, = k=l
P ~ =D lim P-02
number of erroneous deliveries for P packets total number of deliveries for P packets
~,(k).
k=l
The remaining variables, (T, ( been defined in (6) and (7).
~ 2 . a3. a4,
P,and
P f , have
E. Throughput Efficiency
PC = lini P-CO
number of correct deliveries for P packets total number of deliveries for P packets
Using w = P ( D ~ ) P ( T / D ~ and ) Q ~taking ~ ~ , the sums separately, we have
kP,(k) = P,(1) We consider here the number of deliveries made, rather than the number of packets because multiple deliveries of the same packet are possible. When the preceding expressions are used
+ 2Pc(2)+ 3P,(3) + w
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IEEE TRANSACTIONS ON COMMUNICATlONS, VOL. 41, NO. 1, JANUARY 1993
The second sum is
k=l
Q3
4
M
kp'(k) =
00
(I3)
+
k=l
k=5
=
+ +
5(1 - a 4 ) 2 ( 1 - P ) 2 ( p Q4 (1- Q 4 I 2 ( 1 - PI2
- 2a4p) '
(19)
Then, the average number of transmissions per packet may be written as
Substituting from (7e), we obtain
+
a a 3 4 5
- 4a4)
(1 - Q d-,Y The throughput efficiency 77 =
k-5
cc
1/N is then
given by
1
k=5
F. Spurious Transmissions
Putting these together, we obtain
5603
+ (1
-
P)(l
- a41
+
0&3(0
+ a 4 - 2Q4P) .
(1 - p)2(1- a 4 ) 2
The last sum we need is 00
kPl(k) = 4(1)
+ "(2) + 3Pl(3)+ Pl(4)
k=l
+ Pl(4)(5 (1-
4Q4)
Q4I2
Let us define N l . Q l ,
Q2,
'
and Q3 as follows:
A packet delivered may be transmitted again if the feedback message incurs undetectable errors. That is, the receiver has accepted a packet, but due to errors in the feedback channel, the acknowledgment arrives at the transmitter as a request for retransmission. This unnecessary transmission is called here a spurious transmission, and in this section, an expression is derived for the probability of its occurrence. All that is needed here is to identify the terms in P, ( k ) that correspond to spurious transmissions. When the term O Q ~ Q ; - ~ is subtracted from P,(k), the result is the probability of spurious delivery at the kth attempt. The term removed here, ~ a 3 a ; - represents ~, erroneous delivery without any unnecessary transmissions. We can therefore write y9,(k) = P,(k) - ( ~ a 3 a ; - ~ , which gives the probability that the delivery of a packet at the kth transmission is spurious, i.e., see (22) at the bottom of the page. The result can then be used to derive the average probability of a spurious transmission. As transmission proceeds, some will be legitimate and others will spurious. For P packets, we can count the illegitimate transmissions, and also find the total number of transmissions. The ratio of these two numbers will give the relative frequency of illegitimate transmissions. Given P s p n lim { N s p ( P ) / N ( P )where } P+CO
(17) Q2
= PfP2
+ Sp + Sa2,
(already defined),
(18)
X S P ( P ) = for& ver e nkT her of spurious transmissions ac e s, and N ( P )= average number of transmissions for P packets,
ODUOL AND MORGERA: PERFORMANCE EVALUATlON OF THE GENERALIZED TYPE-I1 HYBRID ARQ
37
A
the average probability of a spurious transmission is given by ~ 5 1
(23) This unwanted transmission is one important consequence of feedback errors in ARQ systems, and one which needs to receive more attention.
The average probability of packet loss is defined as follows:
PL = lim P-03
average number of packets lost in P packets average number of packets transmitted in P packets
= SL
+
.
Pl(k), where SL = k=5
+
+
+
A. The Predictor
4
03
+
+
+
G. The Average Probability of Packet Loss
From this definition, we have
produce an estimate of the noise level in the channel for the transmission slot. This estimate can be sent to the transmitter, where it can be used to determine the starting depth for the next packet to be transmitted. This idea is illustrated in Fig. 5. The output marked E is the number of errors in the received block. Suppose the transmission time for a packet is Tt seconds, and the propagation delay is Tp seconds, with Tp = DpT,. The GH-I1 ARQ method always starts the packet transmission at depth-1. If the channel is in a state suitable for depth-1, then the scheme will require TI = 1(Tp T,) = l ( D p 1)Tt seconds to transmit a packet. With a predictor all the 1 blocks will be transmitted together, giving a total time of T2 = Tp Tt1 = ( D p 1)Tt seconds. I f D p >> 1, the ratio of the throughputs will be T1/T2 = l ( D p l ) / ( D p 1 ) FZ 1. This shows that the predictor would yield a throughput improvement. Another advantage of using a predictor is that while without prediction, the method spends time decoding each depth (starting with the lowest) until the final depth, the method with prediction uses only one decoding; all the intermediate decodings are eliminated.
Pl(k). k=l
Substituting from (8) for Pl(k) gives
The loss of packets is another undesirable consequence of feedback errors in ARQ schemes.
The prediction of a parameter from the observations of a random process is a topic that has received considerable attention in engineering and science; it has even been given attention in such areas as economics and other social sciences. It is not the intention of the present work to duplicate any results that have been obtained on this topic. The aim here is to demonstrate that prediction does provide some performance improvement in the GH-I1 ARQ scheme. The time difference between the receiver and the transmitter is D p packet times. First estimate the mean j2 of the error process, as
111. ARQ SCHEMEWITHA PREDICTOR
Since the channel noise level varies from time to time, it is desirable to have an idea of what it is going to be in the next transmission slot; this would facilitate choosing the appropriate coding parameters. Suppose that for each decoding, the decoder produces the weight of the corresponding error pattern. These weights could then be fed into a predictor to
Suppose the predictor is of order P. Then the prediction 2,-1+~, of e,-l+DP, given e,-l and all the preceding P-1 values, is found as follows: P ea-l+D,
= j2
+
ai(e,-i i=l
- j2).
(26)
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TABLE I
RESULTSFOR BER = lop3, D, = 15, Noisy Feedback Without Prediction throughput efficiency 7 undetected error prob. F'UD packet loss prob. PL spurious delivery prob. Psp
cy
= 0.8
Noisy Feedback With Prediction
Noiseless Feedback Without Prediction
0.80 3.22 x 10-7 5.10 x 10-9 2.01 x 10-8
0.85 2.12 x 10-7 -
0.70 9.31 x 10-7 2.11 x 8.92 x lo-'
The predictor coefficients are then determined to minimize the mean square error E' of the predictor where 7 . 0
r
. .
-
. ,....,I
.
-
e0.e c I 0.6 - 5 : E O . 4 -
.9? O
I
-
e 5 0.2 -
P
P
Markov
Chonn-l
a-0.6
or- l b
P
Markov Channel
Optimizing with respect to each
al,
-lo-or . . .I....i lo-' c
10-0
where ci-l = E{(e,-i - ji)(e,-l - b ) } is the autocovariance of the error process. This set of equations can be solved to find the predictor coefficients a;, i = 1,2, , P. In applications, the values of the autocorrelation function are not known, and have to be estimated from past observations. One way [16], [17] to estimate these is
I v . COMPARISON OF RESULTS Simulations were run on a Markov channel with and without feedback errors. For the noisy feedback, there were two sets of simulations-one with a predictor (A4= 40) and the other without. The simulations were run with D p = 15 and varying The noise model used in bit error rate (BER) from lo4 to the simulations was e , = cye,-l (1 - a)u, where the u,'s are independent and identically distributed random variables. It is shown in [15] that the autocovariance function of this process is a decaying function of the lag. Each un is a binomial
+
-
gives
. . .I....i . . . I .... lo-* lo-' I
bit error
. .
. # J
lo-:
rate, p
(b) Fig. 6. (a) Throughput efficiency. (b) Probability of undetected error.
random variable with parameters N and p where N = 504, and p = BER. The number of packets used was lo4, each 504 b long. The two codes used were the (28,7,10) KM code for adaptive error correction, and a (504,484) code for error detection, obtained by shortening the (1023,1003,5) BCH code. For a bit error rate of the results are as shown in Table I. The rest of the results are given in Figs. 6 and 7, from which it is clear that feedback noise lowers the throughput efficiency, increases the probability of undetected error, and creates the loss of packets. It is also evident that the predictor has improved the performance a great deal.
APPENDIX In this Appendix we give the performance expressions for the GH-I1 ARQ Scheme with noiseless feedback to facilitate comparison with the noisy feedback results already found. A. GH-II ARQ Scheme with Noiseless Feedback
Here we will rely heavily on the work already done in the preceding pages. By proceeding as we did previously, we trace
~
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-11 HYBRID ARQ
1
g 10-10
_II
lo-.
10-0
. .
.
I
...,
lo-'
10-4
bit error
io-'
39
lo-:
rate, p
. . . ,...., .
, ,
,..,., .
,
,
,..
D. Correct Delivery By retracing the steps for finding PUD,we find the probability of correct delivery as Pc = Cr=.=,PC(IC). With p.5 = P(C4/D4) . P ( D i ).n:=lP(Dj+i/Dj), pic = Ct,lPc(IC), and p3 as before, we have PC = C~lPc(IC) =
PIC+ P5/(1 - P3). bit error
rots.
p
( b)
E. Conditional Probabilities
Fig. 7. (a) Probability of spurious delivery. (b) Probability of packet loss.
the events leading to correct delivery, undetected error, etc., and find the first equation at the bottom of the page. Continuing for IC 2 4,we obtain the second equation at the bottom of the page. In general for IC 2 5, we obtain
Let SQ = {error interval corresponding to event Q}. With (1 - a)u, where the error model given by e, = ae,-1 the u,'s are independent and identically distributed random variables, the conditional probability P(T/D1) used in the GH-I1 ARQ scheme can be found as follows. First define the random variable W ~ p ( a )as
+
DP
WDP((Y) =
%+m. aDP-m
(1 - a) m=l
(P(D4/04))"',
IC 2 5, and
Then the conditional probability P(T/D1) is given by
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 1, JANUARY 1993
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where ST - j a D Pis a translation of the interval S T . Similar expressions can be found for the remaining conditional probabilities. The random variable W~p(cy)is not necessarily an integer; it is, however, discrete.
ACKNOWLEDGMENT
We are grateful to the anonymous reviewers for their constructive comments and for bringing to our attention some relevant references. REFERENCES [ l ] H. Krishna and S. D. Morgera, “A new error control scheme for hybrid ARQ schemes,” IEEE Trans. Commun., vol. COM-35, pp. 981-990, Oct. 1987. [2] S. D. Morgera and H. Krishna, Digital Signal Processing-Applications to Communications and Algebraic Coding Theories. New York: Academic, 1989. [3] S. Kallel, “Analysis of type-I1 ARQ scheme with code combining,” IEEE Trans. Commun., vol. 38, pp. 1133-1137, Aug. 1990. [4] S. Kallel and D. Haccoun, “Sequential decoding with ARQ and code combining: A robust hybrid FEC/ARQ system,” IEEE Trans. Commun., VOI.38, pp. 773-780, July 1990. [SI __, “Generalized type-I1 hybrid ARQ scheme using punctured convolutional codes,” IEEE Trans. Commun., vol. 38, pp. 1938-1947, Nov. 1990. [6] H. 0. Burton and D. D. Sullivan, “Generalized burst-trapping codes,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 736-742, Nov. 1971. [7] A. R. K. Sastry and L. N. Kanal, “Hybrid error control using retransmission and generalized burst-trapping codes,” IEEE Trans. Commun., VOI.COM-22, pp. 385-393, Apr. 1976. [8] A. M. Michelson and A. H. Levesque, Error-Control Techniques for Digital Communications. New York: Wiley, 198.5. [9] G. D. Forney, Concatenated Codes. Cambridge, MA: M.I.T. Press, 1966. [lo] T. Kasami, T. Fujiwara, and S. Lin, “A concatenated coding scheme for error control,” IEEE Trans. Commun., vol. COM-34, pp. 481-488, May 1986. [ll] K. Mokrani and S. S. Solimani, “Concatenated codes over fading and dispersive channels,” in Proc. IEEE Int. Conf Commun., lCC’S9, 1989, pp. 1378- 1382, (455.1 -45.5.5). [ 121 G. C. Clark and J. B. Cain, Error-Correction Codes for Digital Communications. New York Plenum, 1983. [13] T. Kl0ve and M. Miller, “The detection of errors after error correction decoding,” IEEE Trans. Commun., vol. COM-32, pp. 511-517, May 198.5. [14] C.H.C. Leung, Y. Kikumoto, and S.A. S~rensen,“The throughput efficiency of the go-back-N ARQ scheme under Markov and related error structures,’’ IEEE Trans. Commun., vol. 36, Feb. 1988. [lS] V. K. Oduol, “Performance evaluation of error control methods with noisy feedback: The generalized type-I1 hybrid ARQ scheme and the selective repeat ARQ scheme on Markov and Gilbert channels.” Ph.D. dissertation, McGill University, Dep. Elec. Eng., 1991. [16] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1986. [17] A. A. Giordano, Least Squares Estimation with Applications to Digital Signal Processing. New York: Wiley, 1985, pp. 63-66.
Vitalice K. Oduol was born on August 23, 19.58 in Siaya, Kenya. He received his pre-university education at Alliance High School in Kenya. In 1981 he was awarded a CIDA scholarship to study electrical engineering at McGill University, MontrCal, Canada, where he received the B.Eng. (Hons.) and M.Eng. degrees in 198.5 and 1987, respectively, both in electrical engineering. In October 1991, he completed the requirements for the Ph.D. degree in electrical engineering at McGill University. While at McGill he twice received the Douglas tutorial scholarship. In the graduate program, he was a research associate and teaching assistant. Since November 1989 he has been with MPB Technologies, Inc., where he has participated in a variety of projects. In particular, he completed a feasibility study of meteor burst communication systems. The conclusions of this were presented at the 1990 IEEE Military Communications Conference (MILCOM 90). Later he conducted an evaluation of the satellite communication systems of the near future, with a view to determining the kind of signal processing that can be done on the satellite to improve the overall performance, taking into account the developments in technology. His research interests include adaptive error control, feedback communica$on, modeling and analysis of communication channels.
Salvatore D. Morgera (S’70- M’72 -SM’84 - F’90) received the B.Sc. degree in physics with Honors and the MSc. and Ph.D. degrees in electrical engineering from Brown University in 1968, 1970, and 1975, respectively. From 1968 to 1978, he was a Senior Research Scientist with Raytheon Company, Submarine Signal Division, Portsmouth, RI, and the Principal Investigator and Technical Director for a number of major R & D programs. During the oeriod 1978-198.5. he was a Professor of Electrical Engineering at Concordia University, Montreal, P.Q., Canada; the Responsible Administrator for the Communications, Circuits, and Systems Group; and a Fellow of the Science College. Since 198.5, he has been an Adjunct Professor at Concordia University; Professor of Electrical Engineering at McGill University, Montreal, P.Q., Canada; Area Representative of the McGill Communications Systems Group; and Director of the Information Networks and Systems Laboratory (INSL). His research interests are multidisciplinary and encompass specialized topics within the areas of information and communications theories, applied stochastic processes, computational complexity, and open distributed processing. Dr. Morgera is an author of the book, Digital Signal Processing: Applications to Communications and Algebraic Coding Theories, Academic Press, 1989. Dr. Morgera has served on Organizing Committees, Technical Program Committees, and as a Session Organizer and Chairman for many IEEE, SIAM, and other conferences in the areas of pattem recognition, signal processing, and communications. He is currently Chairman of the Communications Chapter, Montreal Section IEEE, and, during 1988-1990, held a joint appointment as President of the Engineering Grant Selection Committee and Member of the Program Committee for the Quebec Research Council (Fonds pour la Formation des Chercheurs et ]’Aide a la Recherche, FCAR). He is a Major Project Leader and Member of the Program Committee in the newly formed Canadian Institute for Telecommunications Research (CITR), a Government of Canada Network of Centers of Excellence. Y
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Performance Evaluation of the Generalized Type-I1 Hybrid ARQ Scheme with Noisy Feedback on Markov Channels Vitalice K. Oduol and Salvatore D. Morgera, Fellow, ZEEE
Abstract-By taking a particular look at the generalized TypeI1 hybrid ARQ scheme on Markov channels, this work investigates the effect of feedback channel errors on the performance of ARQ systems. In the analysis of ARQ methods, it is quite often assumed that the feedback channel is error-free. However, taking into account the feedback channel errors provides a more realistic evaluation of the performance. The main contribution of this work is that it demonstrates that it is indeed possible to derive expressions for certain critical performance parameters, such as the throughput efficiency, the probabilities of packet loss, undetected error, and correct delivery. To provide a means of comparison, the paper provides, in the Appendix, a parallel set of expressions under the usual assumption of an error-free feedback channel. By use of simulations, the ARQ system performance is examined under the two cases-noiseless feedback and noisy feedback It is found that feedback channel noise can result in the loss of packets, an increase in the number of undetected errors, and the occurrence of unnecessary transmissions. To enhance the performance of the GH-I1 ARQ scheme, a predictor is used and found to provide remarkable performance improvement-lowering the probability of undetected error, reducing the number of unnecessary transmissions, and increasing the throughput efficiency. The results presented are felt to be important for system design employing ARQ error control methods.
CEC is designed in such a way that its generator matrix can be partitioned into m subblocks, each having dimension (nl x nl). The integer m is called the depth of the code used [l],[ 2 ] .In this paper, we present the scheme with block codes; the use of convolutional codes in similar ARQ systems can be found in the work of Kallel and Haccoun [3]-[5]. On channels where the error bursts (of maximum length B ) are separated by a gap (of minimum length G), generalized burst trapping [6], [7] techniques can be used. These techniques differ markedly from the GH-I1 ARQ scheme described here, in that they require certain statistical properties in the bursts, requirements that are not needed in ARQ systems [8]. For GH-I1 ARQ, the generator matrix G of the error correcting code can be partitioned into subblocks G,, to give G = [GllG21G31 . . . IGm].
For the code CECto be useful for adaptive error correction, it is designed so that the subcode C,, with the generator matrix G(') given by
G(') = [GllG21G31 . . . IG,] I. INTRODUCTION
T
HIS section presents the generalized Type-I1 hybrid ARQ scheme (GH-I1 ARQ), which is a method that employs variable redundancy transmission. The amount of redundancy is increased as the number of retransmission requests increases. So far this method of error control has been analyzed under the assumption of an error-free feedback channel. The presentation given in this work treats both cases-noisy feedback and noiseless feedback.
(1)
z 5 m,
(2)
has minimum distance d, such that d, < d, for 1 5 a < J 5 m. The code C, is the final code CECitself. The information block d is formed from an original message D based on the code CED.The mnl-bit codeword is formed from d based on CEC.This codeword is represented as c = [ClIC2(C31 . . '
Icml
(3)
where each c, corresponds to G,, i = 1 , 2 , . . . , m. The block d can be recovered from c, if and only if G, is invertible. The sequence of transmitted blocks until successful decoding A . The Generalized Type-II ARQ Scheme is c1, c2, c3, . . . ,k.When c, is received, the receiver inverts In the GH-I1 ARQ scheme two codes are used. One is a it and determines if there are any errors in the result. If errors high-rate code CED,which is used for error detection only are found, c, is combined with previously received blocks for (used as the outer code), and the other is a rate-l/m code CEC decoding followed by error detection. If errors are found, a (the inner code), which is used adaptively for error correction. NACK is sent to the transmitter to request the block c,+l, and the procedure continues. If no errors are found, an ACK Paper approved by the Editor for Coding and Applications of the IEEE Communications Society. Manuscript received April 19, 1990; revised Septem- is sent to the transmitter and the next packet is transmitted. ber 6 , 1991. This work was supported by Canada Natural Sciences and With the sequence of blocks above, the receiver performs Engineering Research Council under Grants A0912 and A6824, and Quebec error correction based on the codes C1, C2, . . . , C,, having FCAR under Grant EQ-350. minimum distances d l , da, . . . , d,. With each retransmission, V. K. Oduol is with MPB Technologies, Inc., Pointe Claire, P. Q. H9R 1E9, Canada. therefore, a code of a larger minimum distance is used for error S. D. Morgera is with the Department of Electrical Engineering, McGill correction until the code C, is reached. It is clear from this University, MontrCal, P.Q. description that the type-I1 ARQ scheme is a special case of the IEEE Log Number 9206074. 009&6778/93$03 00 0 1993 IEEE
~
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-I1 HYBRID ARQ
33
BCH
ecode
eedback Channel
Fig. 1
The GH-I1 ARQ scheme.
GH-II ARQ scheme, since it is obtained here by setting m. = 2 . We consider the use of KM codes in the GH-II ARQ scheme [2]. The generator matrix of a KM code is block structured as
M = [MllM2). . . IMm].
(4)
The matrices Gi GY1 are related to Mi as
Fig. 2. State transition from one transmission to the next
11. PERFORMANCE EVALUATION In this section, it is assumed that the error process in the channel can be taken as a Markov chain. Specifically, let The generator matrix G is then formed as G = e k ( N ) be the number of errors in a block of length N bits [ G I I G .~. .~IGm].The code with generator matrix [Gl(G21 at time IC. The sequence el(N),e2(N),e3(N), . . . , is taken . . . IG,] is a depth-i code. For an information vector d, if as a Markov chain. The way in which this is an adequate B(’) = dG,, then d can be recovered from I?(’) by the representation of a real channel is discussed in [14], where inversion process d = B(’)GL1. it is stated that the autocovariance of the process can be A block diagram of the GH-11 ARQ scheme is shown in estimated and used to determine whether a Markov chain Fig. 1, from which it is evident that the method uses code model is appropriate. If, for example, the autocovariance is an concatenation, a topic that has received considerable attention. exponentially decaying function of the lag, the Markov model Code concatenation was first proposed by Forney [9] as a can be used. In the same reference, it is stated that such forms practical technique for implementing a code with a very of the autocovariance function have been found to represent long length and a higher error correcting power. More recent “many random phenomena encountered in practice.” With this treatment of this topic can be found in Kasami and others [lo]. assumption, we can derive the expressions for the probabilThis is further discussed by Mokrani and Solimani [ l l ] , who ities of correct delivery, undetected error, and packet loss. point out that as the error rate in a channel increases, a longer, Expressions for noiseless feedback are found in the Appendix. more powerful code is needed, but that since the longer code Let C k , Dk, and Ek, be the events, that the errors at depth-k may have a greater encodeddecoder complexity, this creates are, respectively, correctable, detectable (but not correctable), doubt as to whether there would be any gain. They suggest (in and undetectable. Further, let T and F be the events, respecagreement with Forney) that code concatenation can provide tively, that the feedback message is received as true or false. the required error correcting power with a lower complexity. A packet transmitted may be retransmitted. The decision to This fact is also stated by Clark and Cain [12]. Kl0ve and retransmit is made after the corresponding feedback message Miller [13] discuss the conditions that minimum distances has been received. The feedback message can be true ( T ) of the codes must satisfy for concatenation to bring about or false (6’). The reception of a packet can result in one of an improvement. Therefore, good codes cannot be arbitrarily three possibilities. The errors may be correctable, detectable, concatenated. We mention in passing that the codes used in or undetectable. Referring to Fig. 2, the packet may end in this work do satisfy the Klove-Miller condition. one of the circled states, C4T, C3T, etc.
34
D
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 1, JANUARY 1993
F I I T~
D2F
] epth-3
k-5
Fig. 3.
j=0
Change of depth for the GH-I1 ARQ scheme.
where A. Correct Delivery
P = P(F/F),
As the transmission proceeds, the depth changes in some cases, while in others it does not change. For example, when D1 is followed by a false feedback ( F ) , the transmission of a new packet begins. The transmission of a packet begins at depth-1. Therefore, there is no depth change when 0 1 is followed by a false feedback ( F ) . If D k is followed by a true feedback ( T ) , however, the depth increases, since the requested transmission is used in a higher depth (Dk+l) upon its arrival. This action of depth change is summarized in the illustration of Fig. 3, which shows the changes until the final depth (depth-4) is reached. The labels on the arrows in Fig. 3 indicate the events causing the changes. From this figure, it can be seen that extension to higher depth systems is quite possible. This can be done by appropriately modifying the transitions from depth4. By following the transitions in Fig. 3, it can be shown [15] that the probability P J k ) of correct delivery at the kth transmission is given by
S = P ( D 1 ) .P ( T / D 1 ) .P ( F / T ) ; = P(D1) . P ( T / D 1 ) . P ( E 4 / T ) and.
P,(k) = where
P(C1)P ( T / C l ) k = l P(Dl)P(T/Dl)P2 k=2 P(Dl)P(T/D1)03aZ k=3 P ( D ~ ) P ( T / D I ) P ~ W ~ , "2- 4, ~
{
(6)
a2 = P ( D Z / T ) P ( T / D Z ) = P ( D z / T ). P ( T / D Z ) P ( D 3 / T ). P ( T / D 3 ) a4 = P ( D 4 / T ) .P ( T / D 4 ) and ,& = P ( C k / T ) .P ( T / C k ) '
a3
'
The computation of the conditional probabilities is described in Section E of the Appendix. B. Incorrect Delivery
Incorrect delivery is the delivery of a packet with undetected errors. By using Fig. 3, the events that lead to incorrect delivery are traced to give the probability P p ( k )of incorrect delivery at the k transmission of a packet as
Pf = P ( D 1 ) . P ( F / D 1 ) . This expression will be used later in determining the average probability of incorrect delivery. For now, we proceed to state a similar expression for the loss of a packet. C. Packet Loss
A condition necessary for the loss of a packet is the detection of errors in packet. This by itself is not sufficient to cause the loss of a packet, but if this is followed by a false feedback message then a packet will be lost if the subsequent feedback message is true. Thus, the loss of a packet occurs whenever the event sequence is [Errors Detected] [False feedback message] [True feedback message]. So in deriving the expressions for the loss of a packet, we will trace the event sequences that end with the sequence DFT. The loss occurs in this case for the following reason. If the feedback message following error detection is false, the transmitter will start the transmission of the next packet instead of a retransmission. The receiver will combine the two transmissions for decoding, and since these transmissions belong to different packets, the decoding will be meaningless. The receiver will not be able to detect the problem, since the equivalent number of errors will be greater than the error detecting power of the code used. A delivery will be made to the user and the transmitter informed accordingly. If this message reaches the transmitter unaltered (true feedback), the transmitter will begin transmitting the next packet. So two things have happened here. A packet has been delivered in error, and another has been lost. The scenario described here is illustrated in Fig. 4. By tracing the event sequences ending with the sequence DFT, the probability P / ( k )of packet loss at the kth transmission is given by
35
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-I1 HYBRID ARQ
Transmitter
Packets
Receover
Subblocks A 4 A3 A2 A I
€34 B3 B2
B1
c4 c3 c2 c1
Fig. 4. The loss of a packet.
in the above definitions, the following sums result:
We are now in a position to derive the expressions for the average probabilities of correct delivery and undetected error. D. Average Probability of Undetected Error and Correct Delivery Suppose we have P packets to transmit. We define the probabilities of undetected error and of correct delivery as the limits
When evaluated, (9) and (10) are obtained as shown at the bottom of the page where w = P ( D l ) P ( T / D i ) a s w Q, 2 =
Pfp
+ so + S n 2 ,
4
S, =
4
P ~ ( I carid ) . S, = k=l
P ~ =D lim P-02
number of erroneous deliveries for P packets total number of deliveries for P packets
~,(k).
k=l
The remaining variables, (T, ( been defined in (6) and (7).
~ 2 . a3. a4,
P,and
P f , have
E. Throughput Efficiency
PC = lini P-CO
number of correct deliveries for P packets total number of deliveries for P packets
Using w = P ( D ~ ) P ( T / D ~ and ) Q ~taking ~ ~ , the sums separately, we have
kP,(k) = P,(1) We consider here the number of deliveries made, rather than the number of packets because multiple deliveries of the same packet are possible. When the preceding expressions are used
+ 2Pc(2)+ 3P,(3) + w
36
IEEE TRANSACTIONS ON COMMUNICATlONS, VOL. 41, NO. 1, JANUARY 1993
The second sum is
k=l
Q3
4
M
kp'(k) =
00
(I3)
+
k=l
k=5
=
+ +
5(1 - a 4 ) 2 ( 1 - P ) 2 ( p Q4 (1- Q 4 I 2 ( 1 - PI2
- 2a4p) '
(19)
Then, the average number of transmissions per packet may be written as
Substituting from (7e), we obtain
+
a a 3 4 5
- 4a4)
(1 - Q d-,Y The throughput efficiency 77 =
k-5
cc
1/N is then
given by
1
k=5
F. Spurious Transmissions
Putting these together, we obtain
5603
+ (1
-
P)(l
- a41
+
0&3(0
+ a 4 - 2Q4P) .
(1 - p)2(1- a 4 ) 2
The last sum we need is 00
kPl(k) = 4(1)
+ "(2) + 3Pl(3)+ Pl(4)
k=l
+ Pl(4)(5 (1-
4Q4)
Q4I2
Let us define N l . Q l ,
Q2,
'
and Q3 as follows:
A packet delivered may be transmitted again if the feedback message incurs undetectable errors. That is, the receiver has accepted a packet, but due to errors in the feedback channel, the acknowledgment arrives at the transmitter as a request for retransmission. This unnecessary transmission is called here a spurious transmission, and in this section, an expression is derived for the probability of its occurrence. All that is needed here is to identify the terms in P, ( k ) that correspond to spurious transmissions. When the term O Q ~ Q ; - ~ is subtracted from P,(k), the result is the probability of spurious delivery at the kth attempt. The term removed here, ~ a 3 a ; - represents ~, erroneous delivery without any unnecessary transmissions. We can therefore write y9,(k) = P,(k) - ( ~ a 3 a ; - ~ , which gives the probability that the delivery of a packet at the kth transmission is spurious, i.e., see (22) at the bottom of the page. The result can then be used to derive the average probability of a spurious transmission. As transmission proceeds, some will be legitimate and others will spurious. For P packets, we can count the illegitimate transmissions, and also find the total number of transmissions. The ratio of these two numbers will give the relative frequency of illegitimate transmissions. Given P s p n lim { N s p ( P ) / N ( P )where } P+CO
(17) Q2
= PfP2
+ Sp + Sa2,
(already defined),
(18)
X S P ( P ) = for& ver e nkT her of spurious transmissions ac e s, and N ( P )= average number of transmissions for P packets,
ODUOL AND MORGERA: PERFORMANCE EVALUATlON OF THE GENERALIZED TYPE-I1 HYBRID ARQ
37
A
the average probability of a spurious transmission is given by ~ 5 1
(23) This unwanted transmission is one important consequence of feedback errors in ARQ systems, and one which needs to receive more attention.
The average probability of packet loss is defined as follows:
PL = lim P-03
average number of packets lost in P packets average number of packets transmitted in P packets
= SL
+
.
Pl(k), where SL = k=5
+
+
+
A. The Predictor
4
03
+
+
+
G. The Average Probability of Packet Loss
From this definition, we have
produce an estimate of the noise level in the channel for the transmission slot. This estimate can be sent to the transmitter, where it can be used to determine the starting depth for the next packet to be transmitted. This idea is illustrated in Fig. 5. The output marked E is the number of errors in the received block. Suppose the transmission time for a packet is Tt seconds, and the propagation delay is Tp seconds, with Tp = DpT,. The GH-I1 ARQ method always starts the packet transmission at depth-1. If the channel is in a state suitable for depth-1, then the scheme will require TI = 1(Tp T,) = l ( D p 1)Tt seconds to transmit a packet. With a predictor all the 1 blocks will be transmitted together, giving a total time of T2 = Tp Tt1 = ( D p 1)Tt seconds. I f D p >> 1, the ratio of the throughputs will be T1/T2 = l ( D p l ) / ( D p 1 ) FZ 1. This shows that the predictor would yield a throughput improvement. Another advantage of using a predictor is that while without prediction, the method spends time decoding each depth (starting with the lowest) until the final depth, the method with prediction uses only one decoding; all the intermediate decodings are eliminated.
Pl(k). k=l
Substituting from (8) for Pl(k) gives
The loss of packets is another undesirable consequence of feedback errors in ARQ schemes.
The prediction of a parameter from the observations of a random process is a topic that has received considerable attention in engineering and science; it has even been given attention in such areas as economics and other social sciences. It is not the intention of the present work to duplicate any results that have been obtained on this topic. The aim here is to demonstrate that prediction does provide some performance improvement in the GH-I1 ARQ scheme. The time difference between the receiver and the transmitter is D p packet times. First estimate the mean j2 of the error process, as
111. ARQ SCHEMEWITHA PREDICTOR
Since the channel noise level varies from time to time, it is desirable to have an idea of what it is going to be in the next transmission slot; this would facilitate choosing the appropriate coding parameters. Suppose that for each decoding, the decoder produces the weight of the corresponding error pattern. These weights could then be fed into a predictor to
Suppose the predictor is of order P. Then the prediction 2,-1+~, of e,-l+DP, given e,-l and all the preceding P-1 values, is found as follows: P ea-l+D,
= j2
+
ai(e,-i i=l
- j2).
(26)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 1, JANUARY 1993
38
TABLE I
RESULTSFOR BER = lop3, D, = 15, Noisy Feedback Without Prediction throughput efficiency 7 undetected error prob. F'UD packet loss prob. PL spurious delivery prob. Psp
cy
= 0.8
Noisy Feedback With Prediction
Noiseless Feedback Without Prediction
0.80 3.22 x 10-7 5.10 x 10-9 2.01 x 10-8
0.85 2.12 x 10-7 -
0.70 9.31 x 10-7 2.11 x 8.92 x lo-'
The predictor coefficients are then determined to minimize the mean square error E' of the predictor where 7 . 0
r
. .
-
. ,....,I
.
-
e0.e c I 0.6 - 5 : E O . 4 -
.9? O
I
-
e 5 0.2 -
P
P
Markov
Chonn-l
a-0.6
or- l b
P
Markov Channel
Optimizing with respect to each
al,
-lo-or . . .I....i lo-' c
10-0
where ci-l = E{(e,-i - ji)(e,-l - b ) } is the autocovariance of the error process. This set of equations can be solved to find the predictor coefficients a;, i = 1,2, , P. In applications, the values of the autocorrelation function are not known, and have to be estimated from past observations. One way [16], [17] to estimate these is
I v . COMPARISON OF RESULTS Simulations were run on a Markov channel with and without feedback errors. For the noisy feedback, there were two sets of simulations-one with a predictor (A4= 40) and the other without. The simulations were run with D p = 15 and varying The noise model used in bit error rate (BER) from lo4 to the simulations was e , = cye,-l (1 - a)u, where the u,'s are independent and identically distributed random variables. It is shown in [15] that the autocovariance function of this process is a decaying function of the lag. Each un is a binomial
+
-
gives
. . .I....i . . . I .... lo-* lo-' I
bit error
. .
. # J
lo-:
rate, p
(b) Fig. 6. (a) Throughput efficiency. (b) Probability of undetected error.
random variable with parameters N and p where N = 504, and p = BER. The number of packets used was lo4, each 504 b long. The two codes used were the (28,7,10) KM code for adaptive error correction, and a (504,484) code for error detection, obtained by shortening the (1023,1003,5) BCH code. For a bit error rate of the results are as shown in Table I. The rest of the results are given in Figs. 6 and 7, from which it is clear that feedback noise lowers the throughput efficiency, increases the probability of undetected error, and creates the loss of packets. It is also evident that the predictor has improved the performance a great deal.
APPENDIX In this Appendix we give the performance expressions for the GH-I1 ARQ Scheme with noiseless feedback to facilitate comparison with the noisy feedback results already found. A. GH-II ARQ Scheme with Noiseless Feedback
Here we will rely heavily on the work already done in the preceding pages. By proceeding as we did previously, we trace
~
ODUOL AND MORGERA: PERFORMANCE EVALUATION OF THE GENERALIZED TYPE-11 HYBRID ARQ
1
g 10-10
_II
lo-.
10-0
. .
.
I
...,
lo-'
10-4
bit error
io-'
39
lo-:
rate, p
. . . ,...., .
, ,
,..,., .
,
,
,..
D. Correct Delivery By retracing the steps for finding PUD,we find the probability of correct delivery as Pc = Cr=.=,PC(IC). With p.5 = P(C4/D4) . P ( D i ).n:=lP(Dj+i/Dj), pic = Ct,lPc(IC), and p3 as before, we have PC = C~lPc(IC) =
PIC+ P5/(1 - P3). bit error
rots.
p
( b)
E. Conditional Probabilities
Fig. 7. (a) Probability of spurious delivery. (b) Probability of packet loss.
the events leading to correct delivery, undetected error, etc., and find the first equation at the bottom of the page. Continuing for IC 2 4,we obtain the second equation at the bottom of the page. In general for IC 2 5, we obtain
Let SQ = {error interval corresponding to event Q}. With (1 - a)u, where the error model given by e, = ae,-1 the u,'s are independent and identically distributed random variables, the conditional probability P(T/D1) used in the GH-I1 ARQ scheme can be found as follows. First define the random variable W ~ p ( a )as
+
DP
WDP((Y) =
%+m. aDP-m
(1 - a) m=l
(P(D4/04))"',
IC 2 5, and
Then the conditional probability P(T/D1) is given by
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 41, NO. 1, JANUARY 1993
40
where ST - j a D Pis a translation of the interval S T . Similar expressions can be found for the remaining conditional probabilities. The random variable W~p(cy)is not necessarily an integer; it is, however, discrete.
ACKNOWLEDGMENT
We are grateful to the anonymous reviewers for their constructive comments and for bringing to our attention some relevant references. REFERENCES [ l ] H. Krishna and S. D. Morgera, “A new error control scheme for hybrid ARQ schemes,” IEEE Trans. Commun., vol. COM-35, pp. 981-990, Oct. 1987. [2] S. D. Morgera and H. Krishna, Digital Signal Processing-Applications to Communications and Algebraic Coding Theories. New York: Academic, 1989. [3] S. Kallel, “Analysis of type-I1 ARQ scheme with code combining,” IEEE Trans. Commun., vol. 38, pp. 1133-1137, Aug. 1990. [4] S. Kallel and D. Haccoun, “Sequential decoding with ARQ and code combining: A robust hybrid FEC/ARQ system,” IEEE Trans. Commun., VOI.38, pp. 773-780, July 1990. [SI __, “Generalized type-I1 hybrid ARQ scheme using punctured convolutional codes,” IEEE Trans. Commun., vol. 38, pp. 1938-1947, Nov. 1990. [6] H. 0. Burton and D. D. Sullivan, “Generalized burst-trapping codes,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 736-742, Nov. 1971. [7] A. R. K. Sastry and L. N. Kanal, “Hybrid error control using retransmission and generalized burst-trapping codes,” IEEE Trans. Commun., VOI.COM-22, pp. 385-393, Apr. 1976. [8] A. M. Michelson and A. H. Levesque, Error-Control Techniques for Digital Communications. New York: Wiley, 198.5. [9] G. D. Forney, Concatenated Codes. Cambridge, MA: M.I.T. Press, 1966. [lo] T. Kasami, T. Fujiwara, and S. Lin, “A concatenated coding scheme for error control,” IEEE Trans. Commun., vol. COM-34, pp. 481-488, May 1986. [ll] K. Mokrani and S. S. Solimani, “Concatenated codes over fading and dispersive channels,” in Proc. IEEE Int. Conf Commun., lCC’S9, 1989, pp. 1378- 1382, (455.1 -45.5.5). [ 121 G. C. Clark and J. B. Cain, Error-Correction Codes for Digital Communications. New York Plenum, 1983. [13] T. Kl0ve and M. Miller, “The detection of errors after error correction decoding,” IEEE Trans. Commun., vol. COM-32, pp. 511-517, May 198.5. [14] C.H.C. Leung, Y. Kikumoto, and S.A. S~rensen,“The throughput efficiency of the go-back-N ARQ scheme under Markov and related error structures,’’ IEEE Trans. Commun., vol. 36, Feb. 1988. [lS] V. K. Oduol, “Performance evaluation of error control methods with noisy feedback: The generalized type-I1 hybrid ARQ scheme and the selective repeat ARQ scheme on Markov and Gilbert channels.” Ph.D. dissertation, McGill University, Dep. Elec. Eng., 1991. [16] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1986. [17] A. A. Giordano, Least Squares Estimation with Applications to Digital Signal Processing. New York: Wiley, 1985, pp. 63-66.
Vitalice K. Oduol was born on August 23, 19.58 in Siaya, Kenya. He received his pre-university education at Alliance High School in Kenya. In 1981 he was awarded a CIDA scholarship to study electrical engineering at McGill University, MontrCal, Canada, where he received the B.Eng. (Hons.) and M.Eng. degrees in 198.5 and 1987, respectively, both in electrical engineering. In October 1991, he completed the requirements for the Ph.D. degree in electrical engineering at McGill University. While at McGill he twice received the Douglas tutorial scholarship. In the graduate program, he was a research associate and teaching assistant. Since November 1989 he has been with MPB Technologies, Inc., where he has participated in a variety of projects. In particular, he completed a feasibility study of meteor burst communication systems. The conclusions of this were presented at the 1990 IEEE Military Communications Conference (MILCOM 90). Later he conducted an evaluation of the satellite communication systems of the near future, with a view to determining the kind of signal processing that can be done on the satellite to improve the overall performance, taking into account the developments in technology. His research interests include adaptive error control, feedback communica$on, modeling and analysis of communication channels.
Salvatore D. Morgera (S’70- M’72 -SM’84 - F’90) received the B.Sc. degree in physics with Honors and the MSc. and Ph.D. degrees in electrical engineering from Brown University in 1968, 1970, and 1975, respectively. From 1968 to 1978, he was a Senior Research Scientist with Raytheon Company, Submarine Signal Division, Portsmouth, RI, and the Principal Investigator and Technical Director for a number of major R & D programs. During the oeriod 1978-198.5. he was a Professor of Electrical Engineering at Concordia University, Montreal, P.Q., Canada; the Responsible Administrator for the Communications, Circuits, and Systems Group; and a Fellow of the Science College. Since 198.5, he has been an Adjunct Professor at Concordia University; Professor of Electrical Engineering at McGill University, Montreal, P.Q., Canada; Area Representative of the McGill Communications Systems Group; and Director of the Information Networks and Systems Laboratory (INSL). His research interests are multidisciplinary and encompass specialized topics within the areas of information and communications theories, applied stochastic processes, computational complexity, and open distributed processing. Dr. Morgera is an author of the book, Digital Signal Processing: Applications to Communications and Algebraic Coding Theories, Academic Press, 1989. Dr. Morgera has served on Organizing Committees, Technical Program Committees, and as a Session Organizer and Chairman for many IEEE, SIAM, and other conferences in the areas of pattem recognition, signal processing, and communications. He is currently Chairman of the Communications Chapter, Montreal Section IEEE, and, during 1988-1990, held a joint appointment as President of the Engineering Grant Selection Committee and Member of the Program Committee for the Quebec Research Council (Fonds pour la Formation des Chercheurs et ]’Aide a la Recherche, FCAR). He is a Major Project Leader and Member of the Program Committee in the newly formed Canadian Institute for Telecommunications Research (CITR), a Government of Canada Network of Centers of Excellence. Y
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